Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | #2682 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1041 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(6,9)(10,12)(13,14)(15,21)(16,20)(17,18)(19,22)(23,28)(24,26)(25,27);(2,3)(14,17)(15,22)(16,24)(27,28);(5,6,8,9)(7,10,11,12)(13,15)(14,19)(16,25)(17,21)(18,22)(20,28)(23,24)(26,27);(1,3,4,2)(13,16,18,24)(14,20,17,26)(15,23,22,25)(19,27,21,28);(5,7,8,11)(6,10,9,12)(13,17,18,14)(15,21,22,19)(16,26,24,20)(23,28,25,27)Polycyclic presentation:
2^8|a^2=fh,c^2=gh,d^2=f,e^2=g,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#44637, #7668826, #7669105, … |
| maximal normal subgroup of | C2×#44637, #7668826, #7669105, … |
| maximal quotient of | #535427, #535754, #536098, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 37625);